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Abstract and Applied Analysis
Volume 2014, Article ID 643640, 13 pages
Research Article

Backstepping Synthesis for Feedback Control of First-Order Hyperbolic PDEs with Spatial-Temporal Actuation

1Laboratory of Information & Control Technology, Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China
2The State Key Laboratory of Industrial Control Technology and Institute of Cyber-Systems & Control, Zhejiang University, Hangzhou 310027, China
3Department of Mathematics, Zhejiang University, Hangzhou 310027, China
4College of Mathematics & Information Science, Henan Normal University, Xinxiang 453007, China

Received 27 March 2014; Revised 30 June 2014; Accepted 10 July 2014; Published 14 August 2014

Academic Editor: Milan Pokorny

Copyright © 2014 Xin Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper deals with the stabilization problem of first-order hyperbolic partial differential equations (PDEs) with spatial-temporal actuation over the full physical domains. We assume that the interior actuator can be decomposed into a product of spatial and temporal components, where the spatial component satisfies a specific ordinary differential equation (ODE). A Volterra integral transformation is used to convert the original system into a simple target system using the backstepping-like procedure. Unlike the classical backstepping techniques for boundary control problems of PDEs, the internal actuation can not eliminate the residual term that causes the instability of the open-loop system. Thus, an additional differential transformation is introduced to transfer the input from the interior of the domain onto the boundary. Then, a feedback control law is designed using the classic backstepping technique which can stabilize the first-order hyperbolic PDE system in a finite time, which can be proved by using the semigroup arguments. The effectiveness of the design is illustrated with some numerical simulations.